Restrictions of m-Wythoff Nim and p-complementary Beatty sequences

Summary

Let m be a positive integer. The game of m-Wythoff Nim (A. S. Fraenkel, 1982) is a well-known extension of Wythoff Nim, also known as Corner the Queen. Its set of P-positions may be represented by a pair of increasing sequences of nonnegative integers. It is well-known that these sequences are so-called complementary homogeneous Beatty sequences, that is they satisfy Beatty’s theorem. For a positive integer p, we generalize the solution of m-Wythoff Nim to a pair of p-complementary—each positive integer occurs exactly p times— homogeneous Beatty sequences a = (a_n) _n∊Z≥0 and b = (b_n) _n∊Z≥0 , which, for all n, satisfies b_n - a_n = mn. By the latter property, we show that a and b are unique among all pairs of nondecreasing p-complementary sequences. We prove that such pairs can be partitioned into p pairs of complementary Beatty sequences. Our main results are that {{a_n; b_n} | n ∊ Z_≥0} represents the solution to three new “p-restrictions” of m-Wythoff Nim—of which one has a blocking maneuver on the rook-type options. C. Kimberling has shown that the solution ofWythoff Nim satisfies the complementary equation x_xn = yn-1. We generalize this formula to a certain “p-complementary equation” satisfied by our pair a and b. We also show that one may obtain our new pair of sequences by three so-called Minimal EXclusive algorithms. We conclude with an appendix by Aviezri Fraenkel.

The combinatorial game of Wythoff Nim [Wythoff 1907] is a so-called (2-player) impartial game played on two piles of tokens. As an addition to the rules of the game of Nim [Bouton 1901/02], where the players alternate in removing any finite number of tokens from precisely one of the piles (at most the whole pile), Wythoff Nim also allows removal of the same number of tokens from both piles. The player who removes the last token wins.